Representations of g+++ and the role of space-time -: art. no. 033

We consider the decomposition of the adjoint and fundamental representations of very extended Kac-Moody algebras G(+++) with respect to their regular A type subalgebra which, in the corresponding non-linear realisation, is associated with gravity. We find that for many very extended algebras almost all the A type representations that occur in the decomposition of the fundamental representations also occur in the adjoint representation of G(+++). In particular, for E-8(+++), this applies to all its fundamental representations. However, there are some important examples, such as A(N-3)(+++), where this is not true and indeed the adjoint representation contains no generator that can be identified with a space-time translation. We comment on the significance of these results for how space-time can occur in the non-linear realisation based on G(+++). Finally we show that there is a correspondence between the A representations that occur in the fundamental representation associated with the very extended node and the adjoint representation of G(+++) which is consistent with the interpretation of the former as charges associated with brane solutions.